Exploiting Locality by Nested Dissection
√ For
Square Root Smoothing and Mapping ( SAM )
Peter Krauthausen, Frank Dellaert, Alexander Kipp
College of Computing
Georgia Institute of Technology, Atlanta, GA
[email protected], [email protected], [email protected]
Technical Report number GIT-GVU-05-26
September 2005
Abstract
The problem of creating a map given only the erroneous odometry and feature measurements and locating the own position in this environment is known in the literature
as the Simultaneous Localization and Mapping (SLAM) problem. In this paper we investigate how a Nested Dissection Ordering scheme can improve the the performance
of a recently proposed Square Root Information Smoothing (SRIS) approach. As the
SRIS does perform smoothing rather than filtering the SLAM problem becomes the
Smoothing and Mapping problem (SAM). The computational complexity of the SRIS
solution is dominated by the cost of transforming a matrix of all measurements into a
square root form through factorization. The factorization of a fully dense measurement
matrix has a cubic complexity in the worst case. We show that the computational complexity for the factorization of typical measurement matrices occurring in the SAM
problem can be bound tighter under reasonable assumptions. Our work is motivated
both from a numerical / linear algebra standpoint as well as by submaps used in EKF
solutions to SLAM.
1
1 INTRODUCTION
2
1 Introduction
In recent years the number of applications involving robots mapping an unknown environment has increased immensely. The problem of creating a map given only erroneous
odometry and feature measurements while locating the own position in this environment
is known in the literature as the Simultaneous Localization and Mapping (SLAM) problem. The SLAM problem is fundamental to most robot applications. Recently it has been
proposed to use Square Root Information Smoothing (SRIS) to solve this problem. As
the SRIS does perform smoothing rather than filtering the SLAM problem will become the
Smoothing and Mapping problem (SAM). In contrast to the solutions involving an Extended
Kalman-Filter (EKF) or its dual, the Extended Information-Filter (EIF), this approach delivers a Maximum Likelihood map. Nevertheless this approach suffers from the fact that
the computational complexity is dominated by the cost of transforming a matrix of all measurements into a square root form through factorization. Thus smoothing in the case of a
completely dense measurement matrix has a cubic complexity in the worst case. Since the
measurement matrix is usually quite sparse we can assume that cubic complexity is a very
pessimistic estimation. Furthermore changing the way the matrix is factorized by ordering
the variables differently can greatly reduce the number of operations needed [8]. In this
paper we explain how different ordering algorithms work and how the structure of the measurement matrix is exploited to reduce the runtime. The Nested Dissection algorithm as a
divide-and-conquer approach lends itself very well to exploiting the locality inherent in the
geometrical nature of the mapping task. Given the properties of the computational complexity of the Nested Dissection orderings we obtain tighter complexity bounds for typical
measurement patterns under reasonable assumptions.
In the first section we show how the SAM process can be phrased as a graphical model
as well as a matrix computation. In Section 3 we will use these two representations of the
problem to describe how cunningly changing the order of variables for elimination can improve the runtime performance of a factorization, thus the SRIS. Standard as well as Nested
Dissection orderings are introduced. In the next section we show how the computational
properties of the Nested Dissection can be used to bound the computational complexity of
SRIS for typical measurement patterns. In contrast to the complexity of O(n 3 ) for a dense
matrix factorization we can obtain a complexity bound of at most O(n log 2 n) for the total
3
fill-in and at most O(n 2 ) for the multiplication count for the factorization of a pre-ordered
matrix. Nevertheless these bounds are linked to high constant factors that need to be taken
into consideration when applying the algorithm. In the following two sections an application on an indoor scenario is given and an analysis of the impact of the constants in the
complexity bounds is performed. We conclude this paper with a comparison of submaps which are widely used in EKF solutions - to Nested Dissection partitions.
2 The Smoothing and Mapping Problem
Recent papers [14, 8] show how the process of a robot moving through an environment and
sensing features can be understood as building a graphical model. The vertices V in the
graph G = (E, V ) represent variables and the edges the relations between the variables.
In our problem the variables are the poses of the trajectory of the robot and the features it
has sensed. Whereas the relations are odometry and feature measurements. Thus we are
interested in obtaining the true positions for the poses and features given our measurements.
2 THE SMOOTHING AND MAPPING PROBLEM
3
Given such a graphical model the robot moving from pose x i to xi+1 and sensing features li
in this step is described by adding vertices for xi and li and edges representing the odometry
and feature measurements xi −→ xi+1 and xi −→ lj to G . Fig. 1 depicts such a graph for
a robot which has moved four steps and sensed features at each time step.
G11
 F21 G22


F32


 1
 H
 1
 H1
 2

H32


H42

G33
F43
H53

G44
J!1
J22
J32












3
J4 
J53
Figure 1: Left, measurement graph of a scenario where a robot moves four steps and senses
features at each step. The vertices are the robot positions (circles) and feature positions
(squares). The edges represent the xi −→ xi+1 and xi −→ lj measurements. Right, measurement matrix corresponding to measurement graph on the left. Note that the Jacobians
are block matrices of different size depending on whether they represent measurements
xi −→ xi+1 or xi −→ lj . Each row represents one measurement whereas each column
represents an unknown robot or feature position.
Following [8] this graphical model is equivalent to a system of linear equations or more
compact a measurement matrix A ∈ Rm×n consisting of the Jacobians for the odometry
measurements and the feature measurements with xi and li as variables. A is full-rank per
definition and fulfills the Strong Hall condition1 . Note that A grows enormously as a block
of rows and a block of columns are added for each new measurement. Yet, the matrix is
very sparse2 .
Using the matrix A to solve the mapping problem means solving the least squares problem given by Eq. 1. Whereas θ is our current estimate of the true underlying position of the
robot poses and the features.
θ∗ = arg min kAθ − bk2
(1)
Solving Eq. 1 can be done by using QR or Cholesky factorization by solving the
normal-equations AT A = AT b [2]. For Cholesky factorization this yields the Algorithm 1
and for QR factorization Algorithm 2.
In the case of QR factorization the RHS is most often calculated by attaching it to A. In
addition Q is often not formed but replaced by a series of Householder reflections [7]. Note
1
A matrix A ∈ Rmxn has the Strong Hall Property (SHP) if every m × k sub-matrix, for 1 ≤ k < n, has at
least k + 1 nonzero rows. In real applications least squares problems almost always fulfill this requirement or
A can be permuted into a form that has the SHP. [18, 30]
2
The block size depends on the size of the Jacobians, thus on the dimensionality of the space used. For
example in a planar environment a robot pose is defined by the 2D-Position and an angle describing the heading.
A feature measurement is defined by bearing and range. Therefore the following sizes could be used. F, G ∈
R3×3 , H ∈ R2×3 and J ∈ R2×2 . The block sizes are fixed. For further description we refer the reader to [8].
2 THE SMOOTHING AND MAPPING PROBLEM
4
Algorithm 1 Solving Eq. using Cholesky Factorization
(1) Form I , AT A = RT R
(2) Solve RT y = AT b
(3) Solve Rθ ∗ = y
that due to the uniqueness of the factorization R ∈ Rn×n is the same triangular matrix for
both factorizations.3
Algorithm 2 Solving Eq. using QR Factorization
R
(1) Form QT A =
0
d
(2) Solve QT b =
e
(3) Solve Rθ = d
The aspect of Alg. 1 and Alg. 2 that interests us most is the computational complexity.
The number of non-zero elements (NNZ) initially in the matrix A and the non-zero elements
in R govern the amount of computation needed for the factorization and back-substitution.
We can not influence the first quantity but we can influence the number of non-zeros z
introduced by the factorization by reordering the columns of the A as we will show in
Section 3. Note that the mere size of the matrix A does not determine the amount of
computation. In the coarse algorithm below we show the complexity of the different steps
of the algorithm [25].
3
Some authors denote the lower triangular matrix of the Cholesky factorization as R. To avoid any confusion
of the reader we keep a uniform notation throughout this paper.
3 MATRIX REORDERING
5
Algorithm 3 Iterative solution for least squares problems, [25]
Given A ∈ Rm×n ,
(1) Reordering Aπ ← A- Complexity depends on the method of choice
π
(2) Symbolic Factorization - Compute Space needed for non-zero elements O(z) Time
Repeat
(3) Numeric Factorization - Compute R: O(z 3 ), if A is dense
(4) Back-substitution O(z) Time
Until Convergence
(5) Backorder solution θ ← θπ
π −1
For a least squares problem this sequence of computation has to be repeated until convergence. Yet our experiments have shown that only a very small number of iterations is
needed until convergence.
3 Matrix Reordering
In this section we will show how non-zeroes are introduced in R. We will introduce column
reordering schemes and show how orderings can reduce the NNZ. As we have seen in
Section 2 the matrix A consists of small block matrices. In this section we will also show
the effect of combining the block matrices into one variable in contrast to considering each
column a separate a variable.
We will start this section by presenting standard ordering algorithms and continue with
Nested Dissection orderings. For the latter we will expose how locality of the underlying
matrix structure is exploited automatically. Before we continue in the next section with an
analysis of the computational complexity of Nested Dissection orderings on several graph
types and for typical measurement patterns.
3.1
How reordering effects the NNZ in R
Reordering the matrix A ∈ Rm×n means that we permute the columns. Let τ : {1, .., n} 7→
{1, .., n}be a bijective function that reorders indices. Then the following matrix
P ∈ Rn×n , Pc =
δi,τ (j)
P ∈ Rm×m , Pr =
δτ (i),j
defines a column permutation and
a row permutation.
3 MATRIX REORDERING
6
Eq. 2 shows that for a Cholesky factorization row permutation is obsolete as it is canceled out in the matrix multiplication. Thus only a column permutation will change the
structure of the AT A and therefore the NNZ of the Cholesky triangle. Note that Eq. 2
also shows that the column permutation actually represents a symmetric permutation of the
information matrix4 .
(Pr APc )T (Pr APc ) = PcT (AT A)Pc
(2)
As the QR factorization operates directly on A, thus Pr APc , the row ordering matters
for a QR factorization. Nevertheless if one uses a multifrontal QR approach all rows with
the same leading non-zero elements will be gathered in the same frontal matrix block. This
gives a rough row ordering as the row blocks are usually fairly small, [9]. Because of this
and since both QR and Cholesky factorization will benefit from a good column reordering
we will only discuss column orderings in the rest of this paper.
So far we have been abstract about how an ordering affects the factorization process. It
is far out of the scope of this paper to provide the reader with a detailed understanding of
the underlying mechanisms. The following paragraph shall give the reader an intuition of
how the factorization works. The interested reader is referred to [25, 18] for a more detailed
explanation.
Let A ∈ Rm×n be a symbolic matrix, AT A its information matrix and let the graph
G be constructed as described in Section 2. Vertices in G represent a variable whereas
edges represent the dependencies between them. An edge between two variables means that
they can be expressed as a linear combination of the other and their respective dependant
variables. Furthermore elimination of the matrix can be understood graph-theoretically
as a recursive application of a function on G that eliminates one variable at a time. The
application of this function will eliminate variables in G until there is only one variable left.
Eliminating a variable means that we express this variable through the variables that it is
a linear combination of. When we remove a variable we thus introduce new dependencies
into the graph as we have to link every depending variable of the variable to be eliminated
with the variable it will from now on be expressed by. Graph-theoretically this means that
we remove a node from the G and then add edges to the reduced graph. We make the set
of nodes adjacent to the removed variable a completely connected subgraph, a clique. In
matrix terms this means that we add non-zeros to the R triangle. Thus the optimal solution
is to find an ordering of the variables for the elimination that will result in the least increase
in dependencies overall. Unfortunately finding the next unknown to be eliminated is an
NP-complete problem [1].
3.2
Standard orderings
As we have shown the problem of finding an optimal ordering is NP-complete. This problem has been known for a long time by the linear algebra and scientific computation community [21, 18, 17, 20]. A lot of heuristic approaches have been developed to tackle this
problem. In this section we will present the two most widely used sparse matrix ordering
algorithms for matrix oriented scientific computation on standard desktop machines. Both
approaches are based on the heuristic of eliminating the least constrained variables of G,
thus AT A. The family of algorithms based on this heuristic is known as the Minimum Degree algorithms (MD).
4
We will not discuss any algorithms for obtaining symmetric permutations as to our knowledge the same
performance can be achieved with orderings presented. We refer the interested reader to [6].
3 MATRIX REORDERING
7
A widely used approximation of the original MD heuristic is to eliminate multiple variables in one call of the elimination function, (MMD). Thus if c is the lowest node degree in
G, all nodes with degree c will be eliminated in the next step. In addition indistinguishable
nodes are eliminated. These nodes do not introduce an additional set of dependencies as
they are subsumed by the set produced by another elimination that will be performed in the
same step. MMD saves time on updating the graph and determining the next elimination
candidates [27].
Another approach is to save time on the computation of the exact degrees when eliminating one or more variables. This is done by collecting nodes into cliques so that the
bookkeeping does not need to be done for each node separately but is rather approximated.
Therefore this algorithm is known as Approximate Minimum Degree (AMD) method[1, 6].
AMD is a widely used ordering. For example this reordering technique is used in Matlab’s
colamd command. In comparison to MMD this method is supposed to be faster though
delivering competitive orderings. To illustrate the effect of these standard orderings the
following series of figures shows an example.
The example scenario consists of a robot walking around a block (Fig. 2(a)) and sensing
landmarks. For simplicity we used only the first 8 poses of this walk for the further example.
Fig. 2(b) shows the measurement matrix and Fig. 2(c) a straightened up graph of Fig. 2 (a).
The NNZ of R for various orderings are given in Fig. 3. In this figure a column-ordering
means that every single column and row of A is considered independent of all others. In
a block-ordering the blocks formed by the Jacobians are reduced to a symbolic unit block.
As already described in [8] it makes a huge difference in the NNZ whether AMD is applied
on the symbolic block structure of A or if every column considered standalone. Apparently
AMD does not perform overly well on the column-ordered A. In contrast MMD works
better on A. We attribute this to the fact that the multiple elimination of indistinguishable
nodes works quite well for matrices that have a block structure. Fig. 3 (f) shows that it
is possible to obtain even better orderings. Only a few manually chosen permutations of
AMD ordering were necessary to decrease the NNZ in R. Summarizing we can confirm the
results in [8] and it can be said that already the standard orderings work very well on the
measurement structure underlying the measurement matrix. For the rest of this paper we
will only work on the symbolic block structure of A.
3.3
Nested Dissection orderings
All ordering techniques presented in Section 3.2 try to find an ordering by repeatedly searching the whole graph for the next elimination candidate. A fundamentally different approach
is to apply the divide-and-conquer paradigm. This is feasible as most of the nodes will only
induce new constraints to a set of originally direct neighbors. Nested Dissection orderings
try to exploit this fact by recursively partitioning the graph and returning a post-fix notation
of the partitioning tree as the ordering. This makes Nested Dissection orderings a natural
choice for exploiting locality in graphs. Algorithm 4 shows how the basic Nested Dissection
works.
3 MATRIX REORDERING
8
(a) Simulated walk with 15 poses
(b) Measurement Matrix for the first 8 poses in (a)
(c) Straightened subgraph of the first 8 poses of (a)
Figure 2: Example Scenario
3 MATRIX REORDERING
9
(a) No Ordering - 1459 NNZ
(b) Column MMD - 555 NZ
(c) Block MMD - 658 NNZ
(d) Column AMD - 626
(e) Block AMD - 552 NNZ
(f) Hand Tweaked - 535 NNZ
Figure 3: Column permutations and their effect on the NNZ in the R triangle
3 MATRIX REORDERING
(a)
10
(b)
Figure 4: Planar Graph and Finite Element Mesh
Algorithm 4 Nested Dissection
Let G = (V, E) be a graph, with a set of vertices V and a set of edges E and|G| = n.
√ √
(1) Partition G into subgraphs A, B and C, with |A| , |B| ≤ 32 n and |C| ≤ 2 2 n
(2) Repeat Step (1) until |A| , |B| ≤ or |A| , |B| = 1
Obtain ordering by putting the binary tree of the recursive partitioning in post-order, with
the nodes of the separating set C last for every triple of sets.
The divide-and-conquer approach will only be efficient if the partition of the graph and
the elimination in the local subgraphs can be performed without adding much computation.
There is a lot of literature about balanced graph partitioning. This is the case especially for
planar graphs or finite element graphs (Fig.4) . These are used in physics and mechanics for
large scale simulations. [26] may serve as a good introduction to graph partitioning and de√
livers fundamental properties of graph partitions. Improvements to this n/f (n) separator
theorem mainly focus on finding the separating subgraphs more efficiently. Approaches to
this include spectral analysis of the adjacency structure [32], using partially applied minimum degree orderings as indicators for good partitions [28], pyramidal coarsening and
refining techniques and non-recursive k-way partitioning methods [21]. Other variants will
stop the recursion at a certain level a coarseness and then order the vertices in the graph
arbitrarily or apply one of the standard ordering algorithm described in Section 3.2 on the
larger subgraphs.
k−way partitioning is popular as one might save time on the recursion as well as partitions that might otherwise not influence each other in the k-way partitioning process interact
as they are not hidden in different parts of the recursion. Improving the efficiency of the
partitioning of non-spectral methods is usually done by coarsening the graph down to a certain number of nodes while preserving the topological features. This allows for an efficient
search of a near optimal graph partitioning. This partition of the coarse graph is then refined
4 COMPUTATIONAL COMPLEXITY
11
in a multilevel un-coarsening process of the shrunk graph.
In all Nested Dissection algorithms the separating subgraphs (separator) are of central
importance. As is obvious in the light of how the fill-in comes into existence in the elimination game the size of the separator is crucial to keeping the NNZ low. The aim is to
maximize the number of nodes that are mutually independent. This can only be achieved
with small separators. Therefore most algorithms mentioned so far will consist of a two
step approach of determining the separators in the graph. In the first step the algorithms
will try to find good areas for a cut that preserve the balance which is very important in the
context of parallelizing the calculations on several computers or robots. In a second step a
refinement algorithm like [22] or [13] will be applied. These algorithms can be understood
as optimized variants of bipartite graph matching algorithms as they try to find the minimal
cut between a set of border nodes of the so far determined subgraphs.
3.4
Comparison of AMD and Nested Dissections orderings for SAM
According to the ordering literature AMD and Nested Dissection are the most widely used
reordering techniques [21, 28, 27, 1]. In this section we compare the performance of the
AMD implementation of [6, 5] to the Nested Dissection implementation of [21] on measurement matrices of the kind presented in 2. The measurement matrices were produced
using simulated block-worlds. The result of a walk around one block can be seen in Fig. 2
(a).
In Fig. 5 (a) we can see the results of a walk straight down a hallway of blocks to the
left and right. At each step the robot has sensed between 8 and 12 features. As the robot
needs 4 steps to pass a block this means that if we walked down a 1,000 blocks we will
obtain a measurement matrix with about 4,000 rows and about 40,000 non zero entries.
Fig. 5 (b) and (c) shows the results for walks in a square world of blocks. The number
of blocks denote the length in blocks of one side of the square. The difference between
the two sub-figures is that in Fig. 5 (b) the robot walked pass the given number of blocks.
In the lower sub-figure the number of blocks the robot passed are square to the number of
blocks forming one side of the square. Thus we take into consideration the square gain in
the number of blocks.
Note the scale of the factorization times. For the straight walk in Fig. 5 (a) and for the
random walk in Fig. 5 (b) are roughly the same. Whereas in Fig. 5 (c) the factorization time
is one order of magnitude bigger. The reason for this increase is on the one hand the length
of the walk and on the other hand the density of the measurement graph. The latter aspect
will be described in more detail in the following sections. Nevertheless the main result of
this chapter is that for the SAM measurement matrices up to the given maximum size and
density the orderings work equally well.
4 Computational Complexity
In this section we will present the complexity bounds already known for Nested Dissection
orderings and show how they can be used to derive the computational complexity of a
factorization of the measurement matrix A of the SAM problem. At first we will introduce
complexity bounds for planar graphs and general classes of graphs. We will then show how
graph partitions can be computed and how the size of the separators is bound by typical
measurement patterns. For these typical measurement pattern we will derive complexity
4 COMPUTATIONAL COMPLEXITY
12
(a) Straight walk with blocks to the left and right
(b) Random walk through a square block world
(c) Long Random walk through a square block
world
Figure 5: Factorization times over number blocks. In the case of (b) and (c) the number of
blocks denotes the number of blocks of one side of the square block world.
4 COMPUTATIONAL COMPLEXITY
13
bounds. Finally we will show how this is applicable in a standard indoor scenario.
4.1
Properties of Nested Dissection orderings
As described in the last section, the Nested Dissection algorithm is based on the ability to
recursively partition graphs into subgraphs of roughly equal size with a very small separating subgraph efficiently. The fundamental work about Nested Dissection orderings [25] is
based on the so-called separator theorem [26] and the f (n)-separator theorem.
S EPARATOR T HEOREM: Let G be any n− planar vertex graph. The vertices of G can
be partitioned into three sets, A, B, C such that no edge joins a vertex in A with a vertex
√ √in
2
B, neither do A nor B contain more than 3 n vertices and C contains no more than 2 n
vertices. This partition can be found in O(n) time.
The Separator theorem is a very restrictive statement as it only holds for planar and
finite element graphs. Typical planar graphs or finite element graphs are depicted in Fig.
4. This class of graphs and meshes is found frequently in scientific simulations [21]. A
criterion for the planarity of a graph is given with the Kuratowski Theorem.
(a) A complete bipartite
graph of two sets of three
vertices
(b) Kuratowski graph
Figure 6: It is sufficient for a graph to have one of the above graphs as a subgraph or being
reducible to one of these, to show non-planarity and vice versa.
K URATOWSKI T HEOREM : A graph is planar if and only if it contains neither a complete bipartite graph on two sets of three vertices, Fig. 6 (a), nor a complete graph on five
vertices, Fig. 6 (b).
A general separator theorem for a given class of graphs S is the f (n)- separator theorem.
f (n)-S EPARATOR T HEOREM: A graph is f (n)- separable if there exist constants α, β
with α < 1, β > 0 such that if G is any n−vertex graph S, the vertices of G can be
partitioned into three sets A, B, C such that no edge joins a vertex in A with a vertex in
B, neither A nor B contains more than αn vertices and C contains no more than βf (n)
vertices.
For all classes of graphs for which a f (n)-theorem holds and a partition of such a kind
can be found fast an efficient divide-and-conquer ordering is possible [26]. This is espe-
4 COMPUTATIONAL COMPLEXITY
14
√
√
cially the case if the graphs are planar and f (n) = n, α = 23 and β = 2. In [25] the
√
n-separator theorem was used to prove computational bounds for the complexity of matrix factorizations of sparse matrices whose matrix structure equals a planar graph.
P LANAR N ESTED D ISSECTION T HEOREM: Let G be any planar graph. Then G has
an elimination ordering which produces a fill-in of size c1 n log n + O(n) and a multiplication count of c2 n3/2 + O(n (log n)2 ), where c1 ≤ 129 and c2 ≤ 4002. Such an ordering
can be found in O(n log n) time.
√
For a wider class of graphs exchanging the n- theorem for the planar graph with the
general f (n)− theorem yields the following complexity bounds [25].
R ELAXED N ESTED D ISSECTION T HEOREM 1: Let S be any class of graphs closed
under the subgraph criterion on which an nσ separator theorem holds for σ > 12 . Then for
any n−vertex graph G in S, there is an elimination ordering with O(n 2σ ) fill-in size and
O(n3σ ) multiplication count.
Depending on which statements we can make about f (n) the resulting complexity
bounds for the factorization will be weaker or tighter. It is also assumed that a partitioning
can be found in O(nη ) with η ≤ f (n). This means that the complexity for finding the
partition ought not to be higher than the complexity for the factorization.
This means that in order to apply a Nested Dissection Theorem for a certain class of
graphs we need to show that the graph can be partitioned according to a separator theorem
and then show that obtaining the partition is a less complex process than computing the
ordering.
4.2
Complexity Bounds for Factorizing the Measurement Matrix
The factorization of a dense matrix has a computational cost of O(n 3 ) when using Cholesky
factorization. The factorization of a sparser matrix requires less computational effort. Yet
determining complexity bounds for sparse matrices is difficult. As we have seen in Section
2 the measurement matrix A is quite sparse due to the way it is constructed. The last section
shows that knowledge of the structure of the matrix can be used to obtain tighter bounds
for the complexity of sparse matrix factorizations. In this section we will show how we can
apply the separator theorems with our measurement matrix A and give tighter complexity
bounds for typical measurement patterns.
Let G be a measurement graph, let the number of poses be x and of the features be l. If
S is a separator of G and xS the number of poses and lS the number of features contained
in S. We require the following assumptions to hold.
1. A sensor has a bounded range.
2. For every partition with S the following holds
xS
lS
=
x
l
= r.
3. The number of poses d seen from each landmark is even over the whole graph.
4. For every partition the most distant poses of a connected trajectory are given.
The example in Fig. 7 shows that in the case that a robot uses a forward looking sensor only
and that the robot never walks along an area twice the measurement graph will be planar.
4 COMPUTATIONAL COMPLEXITY
15
This assumes that the forward looking sensor is bound in range such that from pose x i−1 it
will at maximum sense the features that xi will sense that are closest to xi−1 .
Figure 7: Measurement graph for a robot walking three steps and sensing landmarks to both
sides. The circles depict poses and the squares features.
Nevertheless in the general case measurement scenarios do not correspond to planar
measurement graphs. Fig. 8 depicts a small measurement scenario for a not tightly range
restricted omni-directional sensor.
Figure 8: Example of a non-planar measurement graph - the left graph shows the graph Fig.
16(a) with squares denoting features l and circles denoting robot poses x. The original edges
represent only x to l measurements. The dotted lines that were added represent odometry
measurements between the poses. The right graph is isomorph to (a) but laid out to show
the trajectory. From each pose three features are sensed.
The denser the set of measurement will become the less planar our graph will be. In
the rest of this section we will show that it is possible to obtain graph partitions efficiently
for typical indoor measurement scenarios. Thus under reasonable assumptions we can provide a partitioning algorithm comparable to the algorithm in [26] allowing us to apply the
f (n)-separator theorem. We will now give upper bounds for the cardinality of separators
of typical measurement graphs.
S EPARATOR S IZE FOR O NE -WAY WALK : Let G be the measurement graph of a robot
sensing omni-directionally with an even radius of its sensor and bounded in range. If a
robot walks straight and never returns to an already visited environment it is possible to
find a partition of the graph into subgraphs A, B, C such that C contains no more than
2d + 1 vertices.
To prove the above statement consider a straight walk of a robot as in Fig. 9. In worst
case the robot senses in all directions and detects features to the both sides. If we want to
4 COMPUTATIONAL COMPLEXITY
16
Figure 9: Edge cut of a measurement graph with average landmark node degree of 3.
partition the graph by simply placing one cut as shown in the Fig. 9 we cut on average 2d of
the x − l measurement edges and one odometry edge. Making this edge separator a vertex
separator means finding all nodes on one side of the cut who have one edge cut.
S EPARATOR S IZE FOR k-WAY WALK : Let G be the measurement graph of a robot
sensing omni-directionally with an even radius of its sensor and bounded in range. If a
robot walks straight and returns k- times to an already visited environment it is possible
to find a partition of the graph into subgraphs A, B, C such that C contains no more than
(2d + 1)k vertices .
Figure 10: Edge cut of a measurement graph with average landmark node degree of 3 where
the robot saw the environment twice.
The argument for the k-Way statement is analogue to the One-Way statement but we
now have to find the separator over k-trajectories as shown in Fig. 10.
S EPARATOR S IZE FOR k-C ROSSING : Let G be the measurement graph of a robot
sensing omni-directionally with an even radius of its sensor and bounded in range. If a
robot walks k-times over a crossing such that features are seen in common it is possible
to find a partition of the graph into subgraphs A, B, C such that C contains no more than
(2d + 1)k vertices .
Consider two straight walks crossing. As the sensor is bounded in range we know that
4 COMPUTATIONAL COMPLEXITY
17
Figure 11: Edge cut of a measurement graph with a crossing that is visited k-times and
features are seen in common.
on average at the crossing point the robot of the first walk will have seen 2d features. To cut
this part of the robot trajectory we add one vertex. From the crossing c features have been
seen from as much as 2 poses of the crossing trajectory. Whereas 1 ≤ c ≤ 4d. Note that
these features need not be identical with the already observed features as features usually
are not equally good visible from all sides. Again we cut the robot trajectory and therefore
add a vertex. Thus we now have a separator size of (2d + 1)(4d + 1) ≤ (4d + 1)2. For
every further crossing in this point we can assume that at maximum another 4d + 1 vertices
have to be added to the separator yielding a size of (4d + 1)k after k crossings.
Figure 12: Edge cut of a measurement graph with average landmark node degree of 2 where
the robot walks a loop .
S EPARATOR S IZE FOR O NE -k-L OOP : Let G be the measurement graph of a robot
sensing omni-directionally with an even radius of its sensor and bounded in range. If a
robot walks a k-times in a loop it is possible to find a partition of the graph into subgraphs
A, B, C such that C contains no more than (2d + 1)2k vertices .
The statement follows direct from the separator size for a k-way walk. An example is
depicted in Fig. 12. The general case where m loops have to be cut is given as follows.
4 COMPUTATIONAL COMPLEXITY
18
Figure 13: Edge cut of a measurement graph with average landmark node degree of 2 where
the robot walks two loops.
S EPARATOR S IZE FOR M -L OOPS : Let G be the measurement graph of a robot sensing omni-directionally with an even radius of its sensor and bounded in range. If a robot
walks in m loops where the robot maximally sees an environment k-times it is possible to
find a partition of the graph into subgraphs A, B, C such that C contains no more than
(2d + 1)k(m + 1) vertices .
This statement is a result of the last statements. We want to draw the readers attention
to one very useful fact about separator sizes in conjunction with separator theorems. The
theorems only demand that the two large subgraphs have a size of αn with α < 1. In case
xS
lS is high for less than (1−α) nodes we might shift the cut without violation of the balance
restriction and obtain a smaller separator. But note that we might pay for this with a higher
α constant which results in higher constants for the complexity bounds.
So far we have shown upper bounds for the separator sizes of several standard measurement patterns. What remains to be shown is that one can find these separators in a
reasonable number of steps. For all measurement patterns except for the M -loops we can
find roughly the point of the cut as follows. Let us assume cS is the size of the separator
for G. Using Assumption 4 and Assumption 2 we might choose one of the endpoints of G
and then start following the trajectory numbering the poses and connected features until we
reach the pose number closest to α |G| − cS = p. Note that all vertices to be numbered
still need to be inside G. In the case that we do not have a connected subgraph we can still
pick an endpoint and start numbering. Either we will be able to number until we reach the
start pose for the separator or a complete disjoint part of G will be part of this subgraph and
we can continue to number from an endpoint of another disjoint part of G. This algorithm
has similarities to region growing algorithms and k-way partitioning [21]. Indeed for the
M -loops we need to apply a more sophisticated algorithm for determining the poses x p .
These poses can be found by coarsening the graph as described in [21], then determining
the cutting points easily and then refining the coarse graph. As shown in the separator size
statements above such a cS can be determined. Nevertheless we still need to show that the
separator itself can be determined exactly. Considering the one-way-walk we can determine
the partition starting with pose xp . The separator will then consist of xp and all features li
that are seen from xp . In regard to the k-way-walk we extend the separator constructed for
the one-way-walk to contain the last numbered poses xk of the k tracks that are parallel.
4 COMPUTATIONAL COMPLEXITY
19
We then add all features lk that are seen from each xk . In regard to a k − crossing we can
take the separator of the k-way-walk and add all the poses x c that are adjacent to features
already in the separator. In addition we add the roughly 4dx c features that are adjacent to
the newly added poses. In case of a k-loop we can start at a random pose in the loop and
then proceed as if it was a k-way-walk. In regard to the M -loops it is obvious that we can
find the separator just as we did with the k-way-walk once the set of the poses x p is known.
In the last paragraph we have shown that it is possible to determine the separators in a
time linear to the number of vertices of G for the measurement patterns given. We believe
that there are more sophisticated ways of finding these partitions. Our experience is that the
software package based on [21] delivers partitions that are suitable for our purposes. Given
that we can obtain partitions in linear time we still need to determine reasonable α and β for
the f (n)−separator theorem. Let G be a measurement graph and S the maximal separator
of G and cS = |S|. Then for every f (n) = nσ in the sense of the f (n)−separator theorem,
β = cσS can be calculated as cS is constant. Given all these assumptions we can make the
following statements about computational complexity. For planar graphs the next statement
√
follows directly from the n−separator theorem.
P LANAR M EASUREMENT G RAPH B OUND : Given that the assumption holds that the
measurement graph is planar the bounds given in the Planar Nested Dissection Theorems
can be obtained. This means that we can factorize the measurement matrix of the mea3
surement graph with a fill-in of size c1 n log n + O(n) and c2 O(n 2 ) multiplications where
c1 ≤ 129 and c2 ≤ 4002.
C ONSTANT M AXIMAL S EPARATOR T HEOREM : Let G be the measurement graph of a
robot sensing omni-directionally with an even radius of its sensor and bound in range. If the
measurement graph resulting from the robot exploration can be separated by a subgraph of
size smaller than a constant cS there exist constants α, β with α < 1, β > 0 such that if G
is any n−vertex graph S, the vertices of G can be partitioned into three sets A, B, C such
that no edge joins a vertex in A with a vertex in B, neither A nor B contains more than αn
vertices and C contains no more than βf (n) vertices.
C ONSTANT M AXIMAL S EPARATOR B OUND : Let G be the measurement graph of a
robot sensing omni-directionally with an even radius of its sensor and bound in range.
Given that α, β can be found in the sense of the Constant Maximal Separator Theorem the
total f ill − in associated with an ordering produced Alg. 4 on G is at most c 1 n log2 n +
3
O(n) and the multiplication count is at most c2 n 2 + O(n (log n)2 ) where
p
√
1
c1 = β 2 ( + 2 α/(1 − α))/log2 (1/α)
2
and
√
√
√
√
1
c2 = β 2 ( + β α(2 + α/(1 + α) + 4α/(1 − α))/(1 − α))/(1 − δ)
6
with
4 COMPUTATIONAL COMPLEXITY
20
3
3
δ = α 2 + (1 − α) 2
The last two statements show that for example under the assumptions made in this
section for a k-way walk in a hallway we can obtain a
1
1
β 2 = maximal separator size = (2d + 1)k) 2 .
(3)
For α = 23 n and f (n) = n 2 we than obtain the same bounds except for the constants
c1 and c2 as if the graph was planar. The change in the constants is of course dramatic for
small n.
1
In this section we have shown that under reasonable assumptions we can obtain tighter
complexity bounds for the factorization of a measurement matrix that adheres to typical
measurement patterns. These results of course only hold for the measurement patterns presented here. In the following section we use a typical indoor scenario to demonstrate the
partitioning process and show the impact of the changed constants.
4.3
Indoor Scenario
To show the performance of the Nested Dissection we apply it in this section to an indoor
grid world. This problem is challenging as there are no “natural” partitions like rooms to
the sides of a hallway. These might save cuts or keep the cuts small.
Figure 14: Grid World Example
Our grid world and possible cuts in case of a valid the even-degree-assumption are
shown in Fig. 14. The cuts correspond to the cuts of the first Depth-First recursion. For a
4 COMPUTATIONAL COMPLEXITY
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
n
25,000
12446
6196
3071
1518
741
362
172
82
36
14
3
√
n
158.1
111.5
78.7
55.4
38.9
27.2
19.0
13.1
9.0
6
3.6
1.7
21
|A| , |B|
12446
6196
3071
1518
741
362
172
82
36
14
3
stop
|C|
108
54
54
36
36
18
18
9
9
9
9
9
√
β n
1643.1
1159.3
818.0
575.9
404.8
282.8
197.5
136.2
94.1
62.3
38.1
18
Table 1: The level of the recursion, number of nodes n in the each partition,
the next partitions and the separator.
√
n,the size of
grid with 25,000 nodes the partitions in Tab. 1 would be achieved. Where possible we divide
the graph into two equal pieces. If this is not possible we choose the bigger partition in the
recursion. Note that the measurement graph contains only parts where the robot passed
along two times at maximum. A feature is seen from 4 poses on average. For σ = 12 , α = 23
p
we determine β following Eq. 3 as β = (2d + 1)k(m + 1) = 10.3. The results in Tab.
1 show that it is clearly possible to dissect a grid graph recursively into small pieces which
can be easily solved. Note that after the eighth partition all nodes become one subgraph and
the recursion stops.
4.4
Constants in the Complexity Bounds
In this section we want to highlight that the constants in the Constant Maximal Separator
Bound given in Section 4.2 are not negligible.
β
√
18
√
36
√
54
√
108
c1
289.2
578.4
867.5
1753.3
c2
13500.9
38177.0
70127.9
198323.0
Table 2: Values for c1 and c2 of the Constant Maximal Separator Bound for varying values
of β.
Note that these numbers
are quite realistic. For example
for the indoor scenario of
p
√
the last section β = (2d + 1)k(m + 1) = 10.3 = 108, thus c1 = 1753.3 and c2 =
198323.
Fig. 15 shows the extent to which the constants influence the complexity bounds. The
4 COMPUTATIONAL COMPLEXITY
22
(a)
(b)
(c)
(d)
(e)
Figure 15: The top two figures show the fill-in for the planar case, using a β 2 of 18, 36, 54
and 108 as well as n3 and n2 . The middle two figures show the multiplication count for the
same β values. The top and middle figures show 0 ≤ n ≤ 1000 versus 0 ≤ n ≤ 5000.
The bottom figure shows the multiplication count for the same beta values but for 0 ≤ n ≤
9000, 000, 000.
5 SUBMAP MOTIVATION
23
fill-in and the multiplication count roughly behave alike. We want to focus on the middle
and bottom figures, the figures of the multiplication count. In all figures we have additionally plotted the n3 and n2 . It is obvious that except for n ≤ 3500 the complexity bounds
derived with the Constant Maximal Separator Bound are tighter than n 3 . Nevertheless Fig.
15 (e) shows that it will take quite either large n or even tighter approximations of the separator size to achieve a complexity lower than n2 for the sparse matrix factorization with the
given bounds.
5 Submap Motivation
The application of SRIS is a fairly new approach to the mapping problem. The most widely
used approach so far is based on using Extended-Kalman-Filters (EKF) or its dual the
Extended-Information-Filter (EIF). Most of the EKF literature is based on [33] and [34].
Whereas [2] can be seen as the fundamental work on EIFs. In this section we will show the
fundamental problems of both approaches and how extensions exploit the locality inherent
in the mapping problem. In addition we will show how Nested Dissection orderings can be
understood in terms of submaps.
An EKF represents the environment in form of a covariance matrix which becomes
denser each time new observations are made. A problem of the EKF is that it describes
new features through their relation to other already sensed features. Over time the number
of unknowns used to express a new unknown becomes large. Thus an update of the matrix
becomes computationally infeasible [8]. The EIF which is the EKFs dual system represents
the environment by an information matrix which is the inverse of the covariance matrix.
The information matrix has the nice property that any non-zero value in the matrix indicates a strong dependency between two features and thus this matrix is relatively sparse.
Whereas the matrix will still become dense over time.
There have been a variety of approaches aimed at reducing the cost of an update in the
EKF by exploiting the locality in the mapping problem through dividing the environment
or map into several disjoint submaps during the filtering process [23, 31]. The aim is to
calculate an update of the environment covariance matrix with constant computational cost
as one only updates a small, neighboring piece of the environment. The submaps approach
has been well studied and is very promising [3]. An interesting constant time implementation is described in [24]. In the rest of this section we will disregard all the problems
that arise when working with submaps such as how to detect a loop when re-entering an
already existing submap, whether it is better to keep a global frame of reference or several
interconnected frames, how to fuse local maps into a global map or how an information
increase / uncertainty decrease in one submap will be propagated to adjacent submaps thus
improving the quality of the overall map [3, 4, 12, 24]. We focus on how to create the
submap partitions of the environment and what properties these submaps have in contrast
to a Nested Dissection partitioning.
First of all it dividing an environment into submaps is a local on-line process whereas
the Nested Dissection partition is a global offline process. This means that in the optimal
case submaps can be understood as an on-line Nested Dissection partitioning of the map.
Optimal means that the submaps partition the measurements evenly and that the separator
structure of the cuts between submaps is minimal in the graph partitioning sense. Note
5 SUBMAP MOTIVATION
24
that if all measurements are considered equally likely we thereby minimize the conditional
independence of the submaps amongst themselves. Of course these two conditions can be
relaxed by allowing a deviation by a small constant in the number of nodes on the cut and in
the submaps. Note that it is a prerequisite for achieving a constant time EKF algorithm that
the submaps are of roughly equal size [24]. Nested Dissection partitions provide balanced
partitions with small separating structure due to their construction [26, 25]. Reciprocally
to our derivation above we can understand that a Nested Dissection ordering as an ordering
in which we work on the lowest level on submaps and then merge the local results hierarchically into one global map. Nevertheless there is a difference between the lowest level
of partitioning in Nested Dissection and the submaps as such. In the Nested Dissection
algorithm the size of the smallest partition is determined by the computation cost of factorizing this partition [25]. Only in theory every subgraph down to the single vertex level
would be partitioned. In contrast the submaps are designed to have exactly equal size. We
want to highlight that it is known that small submaps are preferable over large submaps
not only due to computational issues but also as the error in the odometry measurements
of the robot grows over time starting a new submap resets the error to zero or gives it less
influence on later measurements. In addition there are results that lead to the conclusion
that submaps might provide better maps if they were aligned and cut according to environment they work on [19, 15]. As proposed in [19, 15] for indoor environments it may make
sense to let whole rooms be one submap so that only the hallway needs to be partitioned.
This definitely would have the advantage that for these rooms no submaps would need to be
connected and no information would need to be propagated through several submaps. This
would mean that the solution for a room will be independent of the rest of the map given
the separator. This would imply that we can solve smaller submaps and then fuse them on
higher level. Hierarchically computing results is well studied in the scientific computation
community [18, 30] as it is a fundamental problem for parallelizing calculations and to the
SLAM community [16, 9]. The locality inherent in the mapping problem is the basis for
every hierarchical mapping algorithm.
As explained earlier in this paragraph the EIF is dual to the EKF and represents the
world in the form of an inverse of the covariance matrix, the information matrix [35]. The
non-zeroes in the information matrix represent the information of the features relative to
each other. Therefore the topological relations between the features is quite obvious from
the entries in the information matrix. There is a strong relation between features if they are
geometrically close. This influence decays rapidly with increasing distance. In [29, 10, 11]
it has been shown that a lot of entries with a value that tends to be zero are negligible and
hardly contribute to the quality of the map. Thus it is claimed that the position of a feature
or the robot can be calculated solely by the elements directly connected with it which form
the so called Markov blanket. Following the approach in [29, 10, 11] newly added features
are made independent from already existing features in the map that are not contained in
the Markov blanket of the new feature, thus the information matrix becomes sparser. This
of course greatly reduces the run time, creating what is called a Sparse EIF (SEIF). Again,
the sparse matrix can be interpreted as graph. Note that if all the inter feature edges in this
information graph are removed and the filtered relations between the poses and the features
were added the information graph would be isomorph to the measurement graph introduced
in [8]. We understand that the sparsification process can be seen as a sliding submap.
We think that the Nested Dissection partitioning could be improved by using one of the
ideas behind the SEIF. We might obtain better separator structures if we not only use the
6 CONCLUSION
25
separator size as a criterion but find a cut that will maximize the intra submap information
and minimize the inter submap information. Whether this separator refinement is applicable will of course depend on the size of the partitions.
After exploring the similarities between EKF submaps and Nested Dissection it is obvious to ask whether Nested Dissection or graph partitioning in general are able to show us
how to construct optimal computational subdivisions for the mapping problem. We think
that this is possible only to a limited degree. The reason is that the recursion of the partitioning process in Nested Dissection will always stop at a certain level or will continue until
there is only one vertex in every subgraph. Thus either the partition is determined by the
complexity of the factorization or there is no coarser granularity of the computational units
than the vertex level. Nevertheless both exploit the locality in the same manor. Whereas
the SEIF exploits the locality not by partitioning but by restricting the mutually influence
of vertices locally but never splits the measurements up into divisions. Whether the information theoretic background of the SEIF can be used to improve the Nested Dissection
partitioning algorithm remains an open question.
6 Conclusion
In this paper we have shown that the smoothing and mapping problem when solved by
a SRIS can be understood as a factorization measurement matrix or a manipulation of a
graph of measurements. On the dual representations we have shown how an ordering of
variables changes the behavior of factorization or respectively the elimination process. Furthermore we have described the properties and complexity bounds on the Nested Dissection
algorithm. In brief the contributions of this paper can be summarized in the following enumeration
• For the SAM problem we have shown that the two most prominent reordering techniques yield comparable results. We have explained how the Nested Dissection algorithm exploits the locality inherent in the SAM problem.
• We give tighter computational complexity bounds for factorizing the measurement
√
matrix. We have shown that with certain bounds on the separator sizes a n −
separator theorem can be derived. This means that in contrast to the complexity of O(n3 ) for a dense matrix factorization we can obtain a complexity bound of
3
O(n log2 n) for the total fill-in and O(n 2 ) for the multiplication count of the factorization of the pre-ordered matrix . Nevertheless these bounds are linked to high
constant factors that need to be taken into consideration when applying the algorithm
and give an example for Nested Dissection in a typical indoor scenario.
• We have exposed the relationship between submaps as used with an EKF and Nested
Dissection. We believe Nested Dissection algorithm might help understand how efficient computational units can be locally determined. We understand that [15] hints
in the same direction.
7 APPENDIX
26
Although the way sparsification is performed in the SEIF we think that a further occupation
with this technique might be beneficial in the development of separator refinement techniques. As we have described in Section 5, the covariance matrix represents features by
their relation to other features already sensed. It remains to be exposed how this filtering
process is related to the Gaussian Elimination process as described in Section 3 and whether
Gaussian Elimination can be understood as a subset of filtering in the Markov Random Field
or vice versa.
Despite our work there is still only very little known about Nested Dissection of measurement graphs. The aim must be a unified complexity bound for the factorization of general measurement graphs given certain assumptions. We think that such a theorem must be
thoroughly examined in regard to how the theoretical bounds reflect the calculation needed
for real world scenarios. Several aspects lend themselves to further investigation such as
what would be the optimal partition size in regard to performance of the cache hierarchy
or how distributed Nested Dissection for swarms of robots exploring an environment can
possibly be developed.
7 Appendix
This is an additional comprehensive summary of the theorems used in this paper and taken
from [25, 26].
K URATOWSKI T HEOREM : A graph is planar if and only if it contains neither a complete bipartite graph on two sets of three vertices, Fig.16(a), nor a complete graph on five
vertices, Fig.16(b).
(a) A complete bipartite
graph of two sets of three
vertices
(b) Kuratowski graph
Figure 16: It is sufficient for a graph to have one of the above graphs as a subgraph or being
reducible to one of these, to show non-planarity and vice versa.
S EPARATOR T HEOREM: Let G be any n− planar vertex graph. The vertices of G can
be partitioned into three sets, A, B, C such that no edge joins a vertex in A with a vertex
√ √in
2
B, neither do A nor B contain more than 3 n vertices and C contains no more than 2 n
vertices. This partition can be found in O(n) time.
f (n)-S EPARATOR T HEOREM: A graph is f (n)- separable if there exist constants α, β
with α < 1, β > 0 such that if G is any n−vertex graph S, the vertices of G can be
partitioned into three sets A, B, C such that no edge joins a vertex in A with a vertex in
7 APPENDIX
27
B, neither A nor B contains more than αn vertices and C contains no more than βf (n)
vertices.
P LANAR N ESTED D ISSECTION T HEOREM: Let G be any planar graph. Then G has
an elimination ordering which produces a fill-in of size c1 n log n + O(n) and a multiplication count of c2 n3/2 + O(n (log n)2 ), where c1 ≤ 129 and c2 ≤ 4002. Such an ordering
can be found in O(n log n) time.
R ELAXED N ESTED D ISSECTION T HEOREM 1: Let S be any class of graphs closed
under the subgraph criterion on which an nσ separator theorem holds for σ > 12 . Then for
any n−vertex graph G in S, there is an elimination ordering with O(n 2σ ) fill-in size and
O(n3σ ) multiplication count.
T IGHTER N ESTED D ISSECTION T HEOREM 1: Let S be any class of graphs closed
under the subgraph criterion on which an nσ separator theorem holds for 13 < σ < 12 . Then
for any n−vertex graph G in S, there is an elimination ordering with O(n) fill-in size and
O(n3σ ) multiplication count.
T IGHTER N ESTED D ISSECTION T HEOREM 2: Let S be any class of graphs closed
√
under the subgraph criterion on which a 3 n−separator theorem holds. Then for any
n−vertex graph G in S, there is an elimination ordering with O(n) fill-in size and O(n log 2 n)
multiplication count.
T IGHTER N ESTED D ISSECTION T HEOREM 3: Let S be any class of graphs closed
under the subgraph criterion on which a nσ separator theorem holds for σ < 31 . Then for
any n−vertex graph G in S, there is an elimination ordering with O(n) fill-in size and
multiplication count.
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